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VII.

Book V. When, of equimultiples, the multiple of the first exceeds the

multiple of the second, but the multiple of the third does not exceed the multiple of the fourth ; the first to the second is said to have a greater ratio than the third to the fourth.

VIII.
Analogy is a similitude of proportions.

IX.
Analogy, at least, consists of three terms.

X.
When three magnitudes are proportionals, the first has to the
third a duplicate ratio of what it has to the second.

XI.
When four magnitudes are proportional, the first has to the

fourth a triplicate ratio of what it has to the second; and
always one more in order as the proportionals shall be extend-
ed.

XII.
Homologous magnitudes, or magnitudes of a like ratio, are such

whose antecedents are to the antecedents and consequents to
the consequents in the same ratio.

XIII. Alternate ratio is the comparing the antecedent with the antecedent, and consequent with the consequent.

XIV. Inverse ratio is, when the confequentis taken as the antecedent, and compared with the antecedent as a consequent.

XV. Compounded ratio is, when the antecedent and consequent, taken as one, are compared with the consequent itself.

XVI. Divided ratio is, when the excess, by which the antecedent exceeds the consequent, is compared with the consequent.

XVII.
Converse ratio is, when the antecedent is compared with the ex-
cess by which the antecedent exceeds the consequent.

XVIII.
Ratio of equality is when there are taken more than two mag-

nitudes in one order, and a like number of magnitudes in ano-
ther order, comparing two to two, being in the same ratio ;
it shall be in the first order of magnitudes, as the first is to the
laft; fo, in the second order of magnitudes, is the first to the
last.

XIX. Ordinate proportion is, the ratio being, as in the last, as the antecedent is to the confequent, in the first order of magnitudes;

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Book V.

so is the antecedent to the consequent in the second order of
magnitudes ; and as the consequent is to any other, so is the
consequent to any other.

XX.
Perturbate proportion is, when there are three or more magni-

tudes, and others equal to them in number, taken two and
two in the same ratio ; in the first order of magnitudes, as the
antecedent is to the consequent; so, in the second order of
magnitudes, is the antecedent to the confequent; and, as in
the first order, the consequent is to some other, so, in the se-
cond order, is fome other to the antecedent.

A X I 0 MS.

I.

E fuldesmare caual?o each of the fame, or of equal magni

tudes, are equal to each other.

II.
These magnitudes that have the fame equimultiples, or whose

equimultiples are equal, are equal to each other.

PRO P. I. THEO R.

F there be any number of magnitudes, equimultiples of a like

number of magnitudes, each of each, whatever multiple any one of the former magnitudes is of its correspondent one, the same multiple are all the former magnitudes of all the latter.

Let AB, CD, be magnitudes, equimultiples of E, F, whatever multiple AB is of E, and CD of F, the fame multiple AB, CD, together, is of E, F, together.

For, let the magnitudes in AB, equal to E, be AG, GB; and the magnitudes in CD, equal to F, be CH, HD, then AG, CH, are equal to E, F; and BG, HD, likewise equal to E, F; therefore, as often as AB contains E, and CD, F, so often AB, CD, contains E, F: Wherefore, if there are, &c.

PRO P. II. THEO R.

IK
F the first be the same multiple of the second, as the third is of

the fourth; and if the fifth be the same multiple of the second, that the fixth is of the fourth; then shall the first, added to the fifth, be the same multiple of the second, that the third, added to the lixth, is of the fourth.

Let the first AB be the same multiple of the second C, that Book V, the third DE is of the fourth F; and let the fifth BG be the fame multiple of the second C, that the fixth EH is of the fourth F; then AG will be the same multiple of C that DH is of F.

For, because AB is the same multiple of C that DE is of F, there are as many magnitudes in AB equal to C, as in DE, equal to F. For the same reason, there are as many magnitudes in BG equal to C, as there are in EH, equal to F; therefore there are as many magnitudes in AG equal to C, as there are in DH equal to Fa Wherefore, &c.

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PRO P. III. THEOR.

IF the first be the same multiple of the second, that the third is of

the fourth, and there be taken equimultiples of the first and third, then will the magnitudes po taken be equimultiples of the second and fourth.

Let the first A be the same multiple of the second B that the third C is of the fourth D; and let EF, GH, be equimultiples of A, C sthen EF is the fame multiple of B, that GH is of D. For, let the magnitudes in EF, equal to A, be FK, KE ; and the magnitudes in GH, equal to C, be HL, LG; then there are as many magnitudes equal to B in FE, as there are magnitudes equal to Ď in GH"; wherefore FE is the same multiple of B, ihat GH is of D. Wherefore, &c.

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PRO P. IV, THE OR.

If the first have the same ratio to the second that the third has to

the fourth, then shall also the equimultiples of the first have the Jame ratio to the equimultiple of the second that the equimultiple of the third has to that of the fourth.

Let there be four magnitudes, A, B, C, D, such, that A is to B as C to D. Let E, F, be taken the same multiples of A, C; and G, H, the same multiples of B, D, then E is to G as F is to H. For, take K, L, any equimultiples of E, F, and M, N, any equimultiples of G, H; then K‘is the same multiple of A' that L is of Ca. For the same reason, M is the a 3: same multiple of B that N is of Do; but, because A is to B as S is to D, if K be equal to M, I will be equal to N; if great

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b def, s.

4 Book V. er, greater, and, if less, less; but K, L, are equimultiples of VE, F, and M, N, of G, H; wherefore E is to Gas F is to

Hb; but it is proved, that, if K be equal to M, L is equal to N; but, if K is equal to M, M is equal to K, and N to L; if greater, greater, and, if less, lefs; wherefore G is to E as H is to F. Wherefore, if four magnitudes be proportional, they will also be inversely proportional. Wherefore, &c.

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IF one magnitude be the same multiple of another magnitude,

that a part taken from the one is of a part taken from the other; then the residue of the one shall be the same multiple of the residue of the other that the whole is of the whole.

Let AB be the same multiple of CD, that a part taken away AE is of a part taken away CF; then the residue EB shall be the same multiple of the residue FD, that the whole AB is of the whole CL.

For, let BE be the same multiple of CG that AE is of CF; then, because AE is the same multiple of CF that BE is of CG, AE will be the same multiple of CF that AB is of GF"; but AE is the same multiple of CF that AB is of CD, and AB is

the fame multiple of GF that it is of CD; therefore GF is e, b Ax. z. qual to CD b; take CF, which is common, from both, there re

mains GC equal to FD; therefore AE is the same multiple of CF that EB is of FD. Wherefore, &c.

PRO P. VI. THEO R.

IF
F two magnitudes be equimultiples of two magnitudes, and fomo

magnitudes, equimultiples of the same, be taken away, the residuc fall either be equal to these magnitudes or equimultiples of them.

For, first, let GB be equal to E ; if HD is not equal to F, let KC be equal to F; then AB is the fame multiple of E that KH is of F a; but AB, CD, are put equimultiples of E, F; therefore

HK is the same multiple of F that CD is of F; therefore KH is Asi zi equal to CD 6. Take CH, which is common, from both, there

remains KC equal to HD; but KC was put equal to F; therefore HD is equal to F; after the same manner it may be demonstrated that, if GB is any equimultiple of E, HD will be the like equimultiple of F. Wherefore, &c.

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